# Temperature and Renormalization Scale in QFT

A particle physicist told me that everything in Peskin & Schroder is at zero temperature, and once you consider finite-$T$ QFT, things become more complicated. Meanwhile, I sometimes see people referring to renormalization scale as temperature. Example: in some theories, there is a grand unification scale at $10^{16}\,\mathrm{Ge\kern-0.08em V}$, and this energy is converted into temperature, and people refer to the time above this temperature in cosmology.

Is finite-$T$ QFT the same as QFT at a higher-energy renormalization scale? Are the concepts related at all?

Short answers: no, they are not the same; they are somewhat related. A more detailed discussion follows.

Indeed, in most QFT books zero temperature is usually assumed. However, if one is interested in energy scales that are way beyond the temperature of the system, the zero-temperature approximation is a valid one. For example, the thermal energy at room temperature is around 25 meV, which means that optical photons are not being thermally created (their energy in vacuum is 1 eV $\ll$ 25 meV), not even talking about thermal generation of electrons (0.5 MeV). Therefore, assuming zero temperature is a very good approximation for, e.g., QED in particle accelerators.

Furthermore, the grand unification scale sets such a high effective temperature that all the physics we know occurs in the effectively zero-temperature limit (with the exception of the early universe, as you have pointed out).

In general, an effective theory at a certain energy scale using a renormalization scheme (renormalization group, RG) is obtained by integrating out all the degrees of freedom higher than that energy scale. In a QFT, the dynamics of these degrees of freedom are determined by, for example, quantum fluctuations and inter-particle interactions. However, a thermal field theory has to do with statistical mechanics, and thus uses the language of ensembles and thermal baths. Moreover, it is possible to perform renormalization procedures for thermal systems as well. In that case, in addition to quantum fluctuations, there are thermal ones. Therefore, the resulting RG equations in general depend on the temperature.

Unfortunately, I do not know of a finite-temperature RG example in vacuum (if that is even well defined). However, there is plenty of examples of RG equations in some medium. One example can be found in Ultracold Quantum Fields by Stoof et al., where in the RG-flow equations (14.64) - (14.66) there are Fermi-distribution functions $N_\uparrow$ that explicitly depend on temperature.

The question how the renormalization group running depends on the temperature is currently still an open question.

The first paper that dealt with this was Renormalization group at finite temperature by H. Matsumoto, Y. Nakano, and H. Umezawa.

To quote from The running coupling at finite temperature by K. Enqvist and K. Kainulainen

There has been many attempts to compute the running coupling constant at finite temperature [1 ], and the literature abounds in contrasting statements [2-5]. Claims for and against asymptotic freedom in high $T$ QCD has both been put forward. The basic difficulty in finding a meaningful definition for the running charge at high temperatures is due to the fact that the coupling constant g depends on two scales, $T$ and an arbitrary renormalization point $\mu$, and it is not obvious how the high T limit should be taken.

So the situation is quite confusing and not clear at all. To get a feeling how confusing the situation is, have a look at https://arxiv.org/abs/hep-ph/9408254.

However, the general problem was nicely summarized in https://arxiv.org/abs/hep-ph/9308227

These problems encountered in finite-temperature field theory arise because one tries to describe one asymptotic regime in terms of effective degrees of freedom associated with another. If one wishes to describe finite-temperature systems, one needs to take into account that the effective degrees of freedom in the system are scale and temperature dependent.

To learn more about this, you could try to search "environmentally friendly renormalization group", which is the name for a research program that tries to make sense of the renormalization group in the presence of external parameters like temperature.